How do I extrude an object into another creating depth?
Select the object you want to use to create the groove, Tab into Edit mode and press E and extrude it to give it some depth (as deep as you want the groove to be). Next select the panel (the white face) and add a Boolean modifier to it selecting the extruded object as the input, change the operation to Difference and then apply the modifier if you are satisfied with the results. Finally, delete the extruded object and you can enter edit mode to tweak as desired.